(\partial)/(\partial x)(y/(x(x+y)))

解

\frac{\partial}{\partial x} (\frac{y}{x ( x + y )})

- \frac{y ( 2 x + y )}{x ^{2} ( x + y ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{y}{x ( x + y )})

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (\frac{1}{x ( x + y )})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= y \frac{\partial}{\partial x} ((x (x + y))^{- 1})

連鎖法則を適用:

- \frac{1}{( x ( x + y ) ) ^{2}} \frac{\partial}{\partial x} (x (x + y))

= - \frac{1}{( x ( x + y ) ) ^{2}} \frac{\partial}{\partial x} (x (x + y))

\frac{\partial}{\partial x} (x (x + y)) = 2 x + y

= y (- \frac{1}{( x ( x + y ) ) ^{2}} (2 x + y))

簡素化

y (- \frac{1}{( x ( x + y ) ) ^{2}} (2 x + y)) : - \frac{y ( 2 x + y )}{x ^{2} ( x + y ) ^{2}}

= - \frac{y ( 2 x + y )}{x ^{2} ( x + y ) ^{2}}